Reaction Order and Ideality Factor in Dye-Sensitized Nanocrystalline

Article pubs.acs.org/JPCC

Reaction Order and Ideality Factor in Dye-Sensitized Nanocrystalline Solar Cells: A Theoretical Investigation M. Ansari-Rad, Y. Abdi, and E. Arzi* Nano-Physics Research Laboratory, Department of Physics, University of Tehran, Tehran, Iran ABSTRACT: Nonlinear recombination in dye-sensitized solar cells was studied from a fundamental point of view. A model based on Marcus theory was used to describe the recombination from both conduction band and trap states. By combination of this model with the empirical form of nonlinear recombination, dependency of the reaction order (β) on the microscopic parameters of the solar cell was investigated. It was analytically shown that β is always less than unity and also depends on the quasi Fermi-level in semiconductors. By this nonconstant β, the dependency of the ideality factor (m), electron diffusion length (Ln), and the electron lifetime (τn) on the Fermi-level were studied. It was discussed that the nonconstant β can explain the flattening of the Ln at high Fermi-level, as observed in some recent experimental works. For the lifetime, it was shown that only the quantity τn/m is accessible in the common open-circuit voltage decay method. It was also shown that the lifetime and the ideality factor can be obtained by the well-known charge extraction method.

1. INTRODUCTION Nanocrystalline dye-sensitized solar cells (DSCs)1 have attracted great attention because of their low cost fabrication process.2 Traditionally, DSCs include a dye-sensitized highly porous (50−60%) TiO2 film electrode, electrolyte containing I−/I3− redox couple filling the pores of the TiO2, and a platinum coated counter electrode. The solar energy to electric power conversion in solar cells is occurred due to electron− hole (e-h) generation, e-h separation, and charge transport.3 In DSCs, e-h pairs are generated via the absorption of light by dye molecules. Photogenerated electrons are then injected into the TiO2 porous network, and the resulting dye cations are reduced by I− ions. Finally, generated I3− ions (holes) in the electrolyte diffuse to the counter electrode, and electrons diffuse from the TiO2 layer to anode.4,5 It has widely been reported that the measured electron diffusion coefficient, Dn, in electrolyte-filled mesoporous TiO2 is several orders of magnitude smaller than that measured in the single-crystal bulk sample.6 In addition, the diffusion coefficient strongly depends on the quasi Fermi-level of the semiconductor, Ef.7−10 A similar behavior, but in the inverse direction, has been observed for electron lifetime, τn, too. For an efficient transport, the diffusion length, Ln ((Dnτn)1/2), must be a few times greater than the film thickness, d (∼10 μm).11 Although in conventional solar cells, the collection efficiency is close to 100% (Ln > d), in the cells fabricated by electrolytes other than I−/I3−, for example, spiro-OMeTAD, τn and Ln are lower as a consequence of high recombination rate of electrons with holes in the electrolyte.12,13 So to optimize the performance of the various types of DSCs, the recombination mechanism, as a key factor of the cell performance, needs to be investigated. The recombination mechanism is still under discussion. In the so-called linear recombination (LR), the rate of back © 2012 American Chemical Society

transfer of electrons from the TiO2 to the electrolyte, is proportional to the electron population in the conduction band, (nc). This assumption leads to have a constant diffusion length Ln for all given Ef, but there are many reports on the variation of Ln versus Ef.12−17 This is often attributed to a nonlinear recombination (NLR) mechanism, in which the back reaction rate is proportional to ncβ, where β is the reaction order.18 It has been suggested that the origin of the NLR is the recombination via surface trap states in the gap region of the semiconductor.19−21 Bisquert et al.19 have provided a sound microscopic view of recombination process using Marcus theory of charge transfer. In the present paper, we combine empirical form of NLR with this theory to obtain the dependency of the reaction order β to the microscopic parameters of the nanostructured semiconductor and the electrolyte in DSCs. It is shown in this paper that β is always less than unity (as reported in many experimental works16,22−24) and also depends on Ef (or open circuit voltage of the cell, Voc). We have shown that β increases up to unity by increasing the Fermi-level. In other words, the system transits from NLR to LR. Dependency of the Ef versus light intensity, ϕ, was also investigated in this study with this nonconstant β. The ideality factor, m, (that is proportional to slope of the semilogarithmic plots of Ef vs ϕ22,23) is then discussed to explain the nonideal behavior in DSCs. Also the β dependence of the lifetime and diffusion length is widely investigated. It is discussed that the nonconstant reaction order can also explain the flattening of the diffusion length at high Ef, as observed in some works.14,16,17 Received: January 31, 2012 Revised: April 21, 2012 Published: April 23, 2012 10867

dx.doi.org/10.1021/jp300984g | J. Phys. Chem. C 2012, 116, 10867−10872

The Journal of Physical Chemistry C

Article

Experimentally, lifetime measurement can be carried out by frequency or time domain methods such as intensity modulated photovoltage spectroscopy (IMVS),25 and stepped lightinduced transient measurement at open circuit condition (OC),26 respectively. These methods are based on the small perturbation approach. The open-circuit voltage decay method (OCVD)27 is also widely used for lifetime measurement. In this method the cell is illuminated under open circuit condition. Then, the light is turned off and Voc is measured with respect to time, t. Since the extracted current is zero, a decrease in Voc can only be attributed to the recombination. Lifetime is then obtained via equation τn = −(kT/q)(dVoc/dt)−1. In this paper we have introduced a generalized equation which can be applied for both LR and NLR. Finally, an equation for lifetime measurement based on the charge extraction method28,29 was proposed in this work that is consistent with results obtained by IMVS and our generalized OCVD methods.

τn =

⎛ E − Ec ⎞ 1 exp⎜ ⎟ kT0 ⎝ kT0 ⎠

1 nc Nc ⎛ nc ⎞ = U= ⎜ ⎟ τc τc ⎝ Nc ⎠

⎛ Ef − Ec ⎞ nc = Nc exp⎜ ⎟ ⎝ kT ⎠

n = Nt

∫ g(E)fFD dE

⎛ n ⎞β U = kcr ⎜ c ⎟ ⎝ Nc ⎠

(7)

where is a constant and β is the reaction order. The NLR rate is often written as U = A(nc)β, in which A is a constant in units of cm−3(1−β) s−1.16,18 Here krc has the unit of cm−3 s−1. Equation 7 may be a more appropriate form of recombination rate, especially when β is not constant. Comparing eq 7 with eq 6 we set krc = Nc/τc (this choice is justified further on). It has been suggested that the NLR is caused by the recombination of the electrons not only via CB states but also via the trap states. Hence we write the NLR rate in a more fundamental form as19,20

(1)

U = Uc + Ut Uc = kcr

(2)

Ut = (2a)

(8)

nc Nc

∫E

Ec

(8a)

k trg (E)fFD dE

f0

(8b)

in which Uc and Ut are the recombination rate via CB and traps, respectively, and Ef 0 is the Fermi-level at dark illumination. Here krt = (Nt/τt(E)) is the electron transfer rate from traps to the electrolyte. It is worth mentioning that τc is constant, but τt depends on the trap energy. By use of Marcus theory (i ≡ c or t)19,35

(2b)

k(ri) = N(i)k(0i)Cox

⎛ (E − E − λ)2 ⎞ (i) f0 kT ⎟ exp⎜⎜ − ⎟ 4kTλ πλ ⎝ ⎠

(9)

where Cox is the concentration of the oxidized species in the electrolyte and λ is the reorganization energy and is a positive quantity. E(i) is equal to Ec for an electron in CB and is equal to the trap energy for an electron in the trap. Also, k0(i) is a constant determined by the amount of electronic coupling between the initial and final states of transferred charge. It must be noted that for a more complete description of the recombination process, incomplete regeneration of the sensitizers can also be considered.36,37 Here we have assumed that the dye molecules are completely regenerated by electrolyte, and therefore we have ignored the possibility of electron recombination with dye cations.38 By use of eqs 7 and 8, the reaction order β can be obtained as

(3)

Here G and U are the rate of charge generation and recombination, respectively, and D0 is the diffusion coefficient of the CB electrons, and is a constant. The Fermi-level dependent effective diffusion coefficient can be obtained from eq331,33,34 −1 ⎛ ∂n ⎞ Dn = D0⎜1 + t ⎟ ∂nc ⎠ ⎝

(6)

krc

where Nc, Nt, and f FD are the effective density of the CB states, volume density of the traps, and the Fermi-Dirac distribution function, respectively. The integral is taken over the gap region. Taking into account traps contribution, continuity equation can be written as16,31 ∂nc ∂n ∂ ⎛ ∂nc ⎞ ⎜D0 ⎟ =G−U− t + ∂t ∂t ∂x ⎝ ∂x ⎠

[LR]

where τc is the lifetime of the CB electrons and is a constant (it is also denoted by τ0). Equation 6 leads to an effective Fermilevel dependent lifetime as τn = τc(1 + (∂nt/∂nc)).31 This results in a constant diffusion length, Ln = (D0τc)1/2 at all Ef, which is not in agreement with experimental reports in which Ln increases when the Fermi-level raises.12−17,23 This observation is mainly attributed to NLR mechanism, in which recombination of the electrons with holes in the electrolyte is given by

where T0 is the characteristic temperature of the traps and Ec is the conduction band energy level. Considering the contribution of the density of electrons in the CB, nc, and the density of electrons in traps, nt, the total electron density, n, in the TiO2 semiconductor can be written as n = nc + nt

(5)

It must be emphasized that n is the total electron density. In LR approximation, the electron recombination rate with holes in the electrolyte is given by

2. THEORY 2.1. Reaction Order. Multiple-trapping (MT) model has been widely used to describe the anomalous dynamics in DSCs.30,31 On the basis of this model, there is a distribution of trap states below the conduction band (CB) of the semiconductor, and electron transport is affected by trappingdetrapping events. The normalized trap energy distribution in this model has an exponential form of30,32 g (E ) =

⎛ ∂U ⎞−1 ⎜ ⎟ ⎝ ∂n ⎠

(4)

In DSCs, lifetime obtained by small perturbation method (such as IMVS method) is defined by13,19,20 10868



(

ln 1 + β=1−

Ut Uc

( )

ln

nc Nc

Article

) (10)

which is always less than unity, as reported in several experimental works.16,22−24 Figure 1 shows the plot of β vs Ef

Figure 2. Semilogarithmic plot of Ef vs recombination rate U for different curves of Figure 1. In region I, recombination occurs mainly via trap states, but in region II recombination via CB dominates and the slopes of curves approaches to 1 × kT, as expected for linear recombination. In the case of c3, the slope has been considered more precisely in Figure 4. −1 ⎧ dVoc kT kT ⎛ nc ⎞ dβ ⎫ kT ⎬ ≡m = ⎨β + ln⎜ ⎟ d ln ϕ q q ⎝ Nc ⎠ dVoc ⎭ q ⎩

Figure 1. Reaction order β vs Fermi level Ef for four different trap characteristic temperatures T0, calculated from eq 10. At low enough Ef, recombination via trap states dominates and β adopts values less than unity. At higher Fermi-level the system transits to linear recombination and β approaches to unity. At a fixed Ef, with the increase of T0, nonlinear recombination becomes more intense. Note that values of the Fermi-level have been expressed relative to Ec = 0. The curves have been named with c1 to c4 for reference in the next figures.

⎪

⎪

⎪

⎪

(11)

In the case of constant β, eq 11 reduces to (dVoc/d ln ϕ) = (kT/βq), and hence m = (1/β). In this case the semilogarithmic curve can be fitted via a line at all Ef. For the case of nonconstant β, the ideality factor is (1/β){1 + (kT/βq) ln(nc/ Nc)(dβ/dVoc)}−1. Figure 3 shows the plot of the ideality factor

for different trap temperatures of 400, 500, 600, and 700 K (the values of constants used for calculations are given in the Appendix). As can be seen in this figure, β is not constant and generally increases by increasing the Fermi level. At high Fermi levels Ut/Uc in eq 10 becomes small, and β approaches to unity. It means that the recombination rate becomes linear with respect to nc at high Fermi levels. On the other hand, at low Fermi levels, β approaches to a value less than unity that depends on the amount of T0. In fact, at higher T0, contribution of the trap states to the recombination increases and nonlinear behavior becomes more intense. It was recently reported that the nonconstant reaction order can be assumed to be possible.16 In addition to T0, the temperature of the nanostructure T, the ratios Nc/Nt and kc0/kt0, and the reorganization energy λ can also change the β behavior. For example with increasing the reorganization energy, contribution of the deep traps to the recombination process decreases and therefore NLR becomes weaker. Experimentally, β can be achieved by semilogarithmic plot of open circuit voltage vs incident light intensity, ϕ. Let us now consider the effect of nonconstant β on the shape of this semilogarithmic plot. Figure 2 shows the plot of Ef vs ln(U) for different β obtained in Figure 1 (at steady state open circuit condition U ∝ ϕ). As shown in this figure, there are two regions at each curve with distinguishable different slopes. The slope of these semilogarithmic plots (that is related to the ideality factor m), can be exactly calculated as below

Figure 3. Ideality factor m vs Fermi level Ef, for different curves of Figure 1. As can be seen the ideality factor is nonconstant and approaches unity at high Fermi level. The maximum in the curves arises from the large (dβ/dEf) when the system transits from nonlinear recombination to the linear one (see eq 11).

vs Ef for different β curves of Figure 1. In this case, semilogarithmic curves can be fitted by two lines with different slopes. Figure 4 shows this fitting for the case of c3 in Figure 1. Aside from recombination via trap states that was considered here, it is shown that the ideality factor greater than unity can also be caused by the back reaction of the electrons via the conductive oxide substrate.39,40 However, by depositing a 10869



Article

Figure 4. Semilogarithmic plot of Ef vs recombination rate U for the curve c3 in Figure 1. The curve has been fitted with two lines, one with the slope of 1.1 × kT (linear recombination regime) and the other with 1.69 × kT (nonlinear recombination regime). These two values can be compared with the precise values in Figure 3 (c3).

Figure 5. Diffusion length Ln vs Fermi level Ef for different curves in Figure 1. As the Fermi level increases, each curve approaches the value (D0τc)1/2, equivalent to the diffusion length for linear recombination. Flattening of the curves at high Ef, is a consequence of the nonconstant reaction order β, as plotted in Figure 1.

blocking compact layer on the substrate, this source of nonideality can often be ignored.23,39,40 These two sources of nonideality (i.e., recombination via trap states or via substrate) can only change the recombination kinetics. It has been discussed that shifting the TiO2 CB with electron concentration41 or a nonideal thermodynamic behavior of CB electrons42 may also cause nonideality. In these cases, nonideality influences both recombination and transport in the solar cells. Consequently, Dn and τn compensate each other completely and Ln becomes constant. Recently, Barnes et al.22 have used simultaneously two different empirical parameters to simulate the DSC: both order parameter of CB electron recombination (equivalent to β in this paper) and the parameter that affects electrons transport. 2.2. Lifetime and Diffusion Length. Values of 0.7−0.8,16 0.59−0.72,23 and 0.8522 have been obtained from the experimental results for β. In these experimental investigations, β has been assumed to be constant. Constant β implies that the diffusion length is a strictly increasing function of the Fermi level. But in some experimental works, it has been observed that the diffusion length increases with Ef, at low and intermediate Ef, and approaches a limit value at high enough Ef.14,16,17 This observation (i.e., the flattening of the diffusion length at high Ef) is consistent with β being nonconstant. Figure 5 plots Ln as a function of Ef for different β curves shown in Figure 1. Ln has been calculated by the expression Ln = (Dnτn)1/2. Dn can be obtained from Eqn.4. For τn, the following equation, based on eqs 5 and 8 has been used19,20

As shown in Figure 5, in all curves, diffusion length increases with Ef and then approaches the limit value Ln = (D0τc)1/2, which is equal to the diffusion length in LR model. In experimental works, 2- to 6-fold increase in diffusion length has been observed when a broad range of Ef (or incident illumination) was scanned.12−17,22−24 From these results and the results of Figure 5 it can be concluded that a β with a little variation over the Ef (c1 and c2 in Figure 1) may be responsible for NLR in the DSCs, because a β with large variation (c4 in Figure 1) can make about a 100-times increase in Ln. It must be noted that a constant β, for example, 0.8, causes a much less increase in Ln than a nonconstant β varying between 0.8 to 1. 2.3. Lifetime Measurement. In the IMVS method25 a small periodic perturbation of the illumination (δϕ) is imposed on the solar cell at open circuit condition, under a background illumination (ϕ), and the corresponding response (δVoc) is measured. δVoc can be written as27 1 δVoc ∝ δϕ ∂U iω + ∂n (13) The characteristic angular frequency (∂U/∂n) is then determined as the inverse of the lifetime at the given Voc. For OCVD method, in the absence of illumination (ϕ = 0) and at open circuit condition (∂nc/∂x = 0), continuity equation can be simplified as

dn = −U dt

⎛ ∂n ⎞ ⎛ ∂U ⎞ ⎟⎟ ·⎜⎜ ⎟⎟ τn = ⎜⎜ ⎝ ∂Ef ⎠ ⎝ ∂Ef ⎠

−1

⎧n = ⎨ c + Nt ⎩ kT ⎪ ⎪

⎧ n ⎨ c + ⎩ kTτc

⎪ ⎪

∫E

Ec v

Combining this with the lifetime definition, eq 5, we can extract the following exact expression for calculating the lifetime (for both LR and NLR) g (E )

⎫ dE⎬ ∂Ef ⎭

∂fFD

⎪

τn−1 = −

⎪

−1 Ec ∂fFD ⎫ r k t g (E ) dE⎬ ∂Ef Ef ⎭ 0

∫

(14)

2

(15)

2

Although (dEf/dt) and (d Ef/dt ) can be obtained from the OCVD measurement, the quantity ((∂2n/∂Ef 2)/(∂n/∂Ef)) is still unknown (that is approximately equal to (1/kT0), unless Ef

⎪ ⎪

−1 dEf ∂ 2n/∂Ef 2 d 2Ef ⎛ dEf ⎞ − · · ⎜ ⎟ dt ∂n/∂Ef dt 2 ⎝ dt ⎠

(12) 10870



Article

→ Ec). One can also extract the following, β-contained, equation for the lifetime −1 ⎛ dEf ⎞−1 ⎧ ⎛ nc ⎞ dβ ⎫ ⎪ ⎪ ⎬ τn = −kT ·⎜ β + · kT ln ⎟ ·⎨ ⎜ ⎟ ⎪ ⎪ ⎝ Nc ⎠ dEf ⎭ ⎝ dt ⎠ ⎩

approximately equal. Also as can be seen, the correction is more important for the systems with high recombination rate via trap states (i.e., at systems with low collection efficiency). Charge extraction method28,29 can be used for measuring lifetime in DSCs, without any assumption about recombination mechanisms. Briefly, the solar cell is illuminated at OC condition until a steady state is reached. The illumination is then turned off and the total electron density, n, is allowed to decay for a given time. Then the cell is short circuited and the extracted charge is measured. The delay time between interruption of the illumination and short circuit charge extraction is scanned to follow the decay of electron concentration. The curves n vs t and n vs Voc are the outputs of this experiment. An equation as τn = −n(dn/dt)−1 has been already used for lifetime measurement.43 But it must be noted that this is not consistent with the lifetime measured from methods such as IMVS or generalized OCVD (i.e., τn = (∂U/ ∂n)−1). With the aid of eqs 5 and 14, from a typical charge extraction technique, lifetime can be calculated via the following equation

(16)

−1

which can be written as τn = −m·(kT/q)(dVoc/dt) , where m is the ideality factor. For β ≫ (dβ/dEf)|Ef − Ec|, it can be obtained that βτn = −

−1 kT ⎛ dVoc ⎞ ⎟ ·⎜ q ⎝ dt ⎠

⎡ ⎤ ⎢β ≫ dβ |Ef − Ec|⎥ ⎢⎣ dEf ⎦⎥

(17)

Consequently, even for β ≫ (dβ/dE f )|E f − E c | (or approximately constant β), only the βτn can be achievable from the common OCVD. For β = 1 (i.e., LR), the common OCVD27 gives the exact value for the lifetime τn = −

−1 kT ⎛ dVoc ⎞ ⎟ ·⎜ q ⎝ dt ⎠

[β = 1]

(18)

dn ⎛ d 2n ⎞ ⎜ ⎟ dt ⎝ dt 2 ⎠

−1

For investigating the difference between the actual lifetime (eq 16) and the lifetime measured by the common OCVD (eq18), we have computed the factor m = {β + kT ln(nc/Nc)(dβ/ dEf)}−1. Figure 6 shows the plot of lifetimes obtained by eq 16 (exact relation for τn) and eq 18 (common OCVD). As can be seen in this figure, the actual lifetime is generally greater than the OCVD one. At high enough Ef, as noted earlier, the main contribution to recombination comes from the conduction band electrons: Ut + Uc ∼ Uc. Hence eqs 16 and 18 are

τn = −

(19)

It is worthwhile noting that since charge extraction method also gives Voc as a function of t, one can obtain the ideality factor m from this method too, as (by combining eqs 16 and 19) −1

m

−1 q dVoc q dVoc dn ⎛ d 2n ⎞ = · ·τn = · · ·⎜ ⎟ kT dt kT dt dt ⎝ dt 2 ⎠

(20)

Since from the semilogarithmic plot of n (∼nt) vs Voc one can also measure the trap characteristic temperature, T0,44 it seems that charge extraction method is a comprehensive technique for getting information about the kinetic parameters of DSCs such as T0, m(β), and τn.

■

SUMMARY AND CONCLUSIONS Nonideal behavior in DSCs was studied in this work, assuming nonideality comes only from nonlinear recombination. Marcus theory was used to model the recombination in both conduction band and trap states. By this model, it was analytically shown that reaction order is always less than unity, and furthermore it is a function of the electron concentration (or Ef). At low Ef, β is less than unity, and when Ef raises, it approaches unity. It means that the system transits from NLR to LR. It was shown that the flattening of the diffusion length at high Ef, as observed in some experimental works, can be reproduced by this nonconstant β. Ideality factor, m, was also investigated, and its dependency on the Ef was studied. It was shown that only the quantity (τn/m) is accessible in the common open-circuit voltage decay method. It was also shown that m and also τn can be obtained from the well-known charge extraction method.

■

APPENDIX Table 1 shows the constants used in our numerical calculations. All integrals were calculated numerically. Integral of eq 2b can be estimated at zero temperature by Nt exp((Ef − Ec)/kT0).31 But for having enough accuracy at high Ef, the integral was calculated numerically, over the gap region, from Ev (valence band edge energy) to Ec.

Figure 6. Comparison between the lifetimes obtained from the common OCDV method (eq 18) and the exact one (eq 16) for curves c1 and c4 of Figure 1. For c1 (weakly nonlinear system) two methods approximately coincide over this Ef range. But for c4 (strongly nonlinear system) the OCVD method gives lower values than the actual ones. 10871



Article

(12) Fabregat-Santiago, F.; Bisquert, J.; Cevey, L.; Chen, P.; Wang, M.; Zakeeruddin, S. M.; Grätzel, M. J. Am. Chem. Soc. 2009, 131, 558− 562. (13) Bisquert, J.; Fabregat-Santiago, F.; Mora-Seró, I.; GarciaBelmonte, G.; Gimenéz, S. J. Phys. Chem. C 2009, 113, 17278−17290. (14) Barnes, P. R. F.; Anderson, A. Y.; Koops, S. E.; Durrant, J. R.; O’Regan, B. C. J. Phys. Chem. C 2009, 113 (3), 1126−1136. (15) Wang, M.; Chen, P.; Humphry-Baker, R.; Zakeeruddin, S. M.; Gratzel, M. Chem. Phys. Chem. 2009, 10, 290−299. (16) Villanueva-Cab, J.; Wang, H.; Oskam, G.; Peter, L. M. J. Phys. Chem. Lett. 2010, 1, 748−751. (17) Chen, J.; Li, B.; Zheng, J.; Jia, S.; Zhao, J.; Jing, H.; Zhu, Z. J. Phys. Chem. C 2011, 115, 7104−7113. (18) Bisquert, J.; Mora-Seró, I. J. Phys. Chem. Lett. 2010, 1, 450−456. (19) Bisquert, J.; Zaban, A.; Greenshtein, M.; Mora-Seró, I. J. Am. Chem. Soc. 2004, 126, 13550−13559. (20) Ondersma, J. W; Hamann, Th. W. J. Am. Chem. Soc. 2011, 133, 8264−8271. (21) Gonzalez-Vazquez, J. P.; Anta, J. A.; Bisquert, J. J. Phys. Chem. C 2010, 114, 8552−8558. (22) Barnes, P. R. F.; Anderson, A. Y.; Durrant, J. R.; O’Regan, B. C. Phys. Chem. Chem. Phys. 2011, 13, 5798−5816. (23) Guillén, E.; Peter, L. M.; Anta., J. A. J. Phys. Chem. C 2011, 115 (45), 22622−22632. (24) Jennings, J. R.; Li, F.; Wang, Q. J. Phys. Chem. C 2010, 114, 14665−14674. (25) Schlichthörl, G.; Huang, S. Y.; Sprague, J.; Frank, A. J. J. Phys. Chem. B 1997, 101, 8141−8155. (26) Nakade, S.; Kanzaki, T.; Wada, Y.; Yanagida, S. Langmuir 2005, 21, 10803−10807. (27) Zaban, A.; Greenshtein, M.; Bisquert, J. Chem. Phys. Chem. 2003, 4, 859−864. (28) Duffy, N. W.; Peter, L. M.; Rajapakse, R. M. G.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 8916−8919. (29) Duffy, N. W.; Peter, L. M.; Rajapakse, R. M. G.; Wijayantha, K. G. U. Electrochem. Commun. 2000, 2, 658−662. (30) Nelson, J. Phys. Rev. B 1999, 59, 15374−15380. (31) Bisquert, J.; Vikhrenko, V. S. J. Phys. Chem. B 2004, 108, 2313− 2322. (32) van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 4292−4294. (33) Anta, J. A.; Mora-Seró, I.; Dittrich, Th.; Bisquert, J. Phys. Chem. Chem. Phys. 2008, 10, 4478−4485. (34) Ansari-Rad, M.; Abdi, Y.; Arzi, E. J. Phys. Chem. C 2012, 116, 3212−3218. (35) Bisquert, J.; Zaban, A.; Salvador, P. J. Phys. Chem. B 2002, 106, 8774−8782. (36) Barnes, P. R. F.; Anderson, A. Y.; Juozapavicius, M.; Liu, L.; Li, X.; Palomares, E.; Forneli, A.; O’Regan, B. C. Phys. Chem. Chem. Phys. 2011, 13, 3547−3558. (37) Jennings, J. R.; Liu, Y.; Wang, Q. J. Phys. Chem. C 2011, 115, 15109−15120. (38) Němec, H.; Rochford, J.; Taratula, O.; Galoppini, E.; Kužel, P.; Polívka, T.; Yartsev, A.; Sundstörm, V. Phys. Rev. Lett. 2010, 104, 197401−4. (39) Cameron, P. J.; Peter, L. M.; Hore, S. J. Phys. Chem. B 2005, 109, 930−936. (40) Peter, L. M. J. Phys. Chem. C 2007, 111, 6601−6612. (41) O’Regan, B. C.; Durrant, J. R. Acc. Chem. Res. 2009, 42, 1799− 1808. (42) Jennings, J. R.; Ghicov, A.; Peter, L. M.; Schmuki, P.; Walker, A. B. J. Am. Chem. Soc. 2008, 130, 13364−13372. (43) Boschloo, G.; Hagfeldt, A. J. Phys. Chem. B 2005, 109, 12093− 12098. (44) Bailes, M.; Cameron, P. G.; Lobato, K.; Peter, L. M. J. Phys. Chem. B 2005, 109, 15429−15435. (45) Anta, J. A.; Mora-Seró, I.; Dittrich, Th.; Bisquert, J. J. Phys. Chem. C 2007, 111, 13997−14000.

Table 1. Parameters Used in Numerical Calculations parameter

value

T (K) Ec (eV) Ev (eV) Ef 0 (eV)

300 0.0 −3.2 −1.0

Nc (cm−3) Nt (cm−3) T0 (K) λ (eV) Cox (cm−3) k0c (cm3/s) f × k0t (cm3/s) D0 (cm2/s)

8 × 1020 4 × 1020 400 − 700 0.65 3 × 1019 1.1 × 1014 1.1 × 1011 0.4

reference

2 16 19,20 19,34,45 34,45 2 19 19 19,20 6,16

Contribution of the trap states to recombination process is mainly from the traps located in the surface of the nanoparticles. In addition, some surface traps may not be distributed upon eq 1 (for example, it may distributed as a Gaussian function20). In this work, for having a physical picture, we simplified our model by assuming the exponential distribution of the trap states, inside the nanoparticles. Also, it was assumed that a fraction f (∼0.2520) of this traps that are located in the vicinity of nanoparticles surface can contribute in recombination.

■

AUTHOR INFORMATION

Corresponding Author

*Phone/Fax: +98 21 61118610. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We would like to thank the Iran National Science Foundation (INSF) for partial financial support. Partial financial support of the “Centre of Excellence on the Structure and Physical Properties of Matter” of the University of Tehran is also acknowledged.

■

REFERENCES

(1) O’Regan, B.; Grätzel, M. Nature 1991, 353, 737−740. (2) Hagfeldt, A.; Boschloo, G.; Sun, L.; Kloo, L.; Pettersson, H. Chem. Rev. 2010, 110, 6595−6663. (3) Würfel, P. Physics of Solar Cells: From Principles to New Concepts; Wiley-VCH: Germany, 2005. (4) Sölbrand, A.; Lindstrom, H.; Hagfeldt, A.; Lindquist, S. E.; Södergren, S. J. Phys. Chem. B 1997, 101, 2514−2518. (5) van de Lagemaat, J.; Park, N. G.; Frank, A. J. J. Phys. Chem. B 2000, 104, 2044−2052. (6) Forro, L.; Chauvet, O.; Emin, D.; Zuppiroli, L.; Berger, H.; Lévy, F. J. J. Appl. Phys. 1994, 75, 633−635. (7) Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021−17027. (8) Sölbrand, A.; Henningsson, A.; Södergren, S.; Lindström, H.; Hagfeldt, A.; Lindquist, S. E. J. Phys. Chem. B 1999, 103, 1078−1083. (9) Frank, A. J.; Kopidakis, N.; van de Lagemaat, J. Coord. Chem. Rev. 2004, 248, 1165−1179. (10) Wang, M.; Chamberland, N.; Breau, L.; Moser, J. E.; HumphryBaker, R.; Marsan, B.; Zakeeruddin, S. M.; Gratzel, M. Nature Chem. 2010, 2, 385−389. (11) Peter, L. M. J. Phys. Chem. Lett. 2011, 2, 1861−1867. 10872


Reaction Order and Ideality Factor in Dye-Sensitized Nanocrystalline

Recommend Documents